T18-1 φ-拓扑量子计算定理 - 形式化规范

类型定义

from typing import Dict, List, Tuple, Optional, Set, FrozenSet
from dataclasses import dataclass
import numpy as np
from enum import Enum
from phi_arithmetic import PhiReal, PhiComplex, PhiMatrix

class TopologicalPhaseType(Enum):
    """拓扑相类型"""
    TRIVIAL = 0      # 平凡相 r=1
    ISING = 1        # Ising相 r=1  
    FIBONACCI = 2    # Fibonacci相 r=2
    TRIPLE = 3       # 三重态相 r=3
    HIGHER = 4       # 高阶相 r>3

@dataclass
class TopologicalPhase:
    """拓扑相"""
    phase_type: TopologicalPhaseType
    topological_rank: int  # 拓扑秩 r_n = F_n（其中n为相位指数）
    energy_gap: PhiReal   # 拓扑能隙 Δ_n = Δ_0 * φ^(-n)（其中n为相位指数）
    coherence_time: PhiReal  # 相干时间 τ = τ_0 * φ^n（其中n为相位指数）
    phase_index: int      # 相位索引
    
    def __post_init__(self):
        """验证拓扑相的一致性"""
        fib_rank = self._fibonacci_number(self.phase_index)
        if self.topological_rank != fib_rank:
            raise ValueError(f"拓扑秩{self.topological_rank}与Fibonacci数{fib_rank}不匹配")
    
    def _fibonacci_number(self, n: int) -> int:
        """计算第n个Fibonacci数（n为输入索引）"""
        if n <= 1:
            return 1
        a, b = 1, 1
        for _ in range(n - 1):
            a, b = b, a + b
        return b
    
    def is_valid_no11(self) -> bool:
        """验证拓扑相是否满足no-11约束"""
        # 拓扑激发模式不能包含相邻的"11"
        binary_rep = format(self.phase_index, 'b')
        return '11' not in binary_rep
    
    def compute_fusion_dimension(self) -> PhiReal:
        """计算融合维度"""
        return PHI ** PhiReal.from_decimal(self.phase_index)

@dataclass
class Anyon:
    """任意子"""
    label: int              # 任意子标签
    statistical_phase: PhiReal  # 统计相位 θ = 2π/φ^|label|
    fusion_rules: Dict[int, PhiReal]  # 融合规则 N_{ab}^c
    topological_charge: PhiReal  # 拓扑荷
    
    def __post_init__(self):
        """初始化统计相位"""
        if self.statistical_phase is None:
            # θ = 2π/φ^|label|
            self.statistical_phase = PhiReal.from_decimal(2 * np.pi) / (PHI ** abs(self.label))
    
    def compute_braiding_phase(self, other: 'Anyon') -> PhiReal:
        """计算与另一个任意子的编织相位"""
        phase_diff = abs(self.label - other.label)
        return PhiReal.from_decimal(2 * np.pi) / (PHI ** phase_diff)
    
    def is_abelian(self) -> bool:
        """判断是否为阿贝尔任意子"""
        return self.label <= 1
    
    def satisfies_no11_constraint(self, other: 'Anyon') -> bool:
        """验证与另一个任意子的组合是否满足no-11约束"""
        return abs(self.label - other.label) != 1

@dataclass 
class BraidingOperation:
    """编织操作"""
    anyon1: Anyon
    anyon2: Anyon
    braiding_matrix: PhiMatrix  # 编织矩阵
    operation_sequence: List[int]  # 操作序列
    
    def __post_init__(self):
        """验证编织操作的有效性"""
        if not self._validate_braiding_sequence():
            raise ValueError("编织序列违反no-11约束")
        
        if self.braiding_matrix is None:
            self.braiding_matrix = self._compute_braiding_matrix()
    
    def _validate_braiding_sequence(self) -> bool:
        """验证编织序列满足no-11约束"""
        seq = self.operation_sequence
        for i in range(len(seq) - 1):
            if seq[i] == seq[i+1]:  # 连续相同操作违反no-11
                return False
        return True
    
    def _compute_braiding_matrix(self) -> PhiMatrix:
        """计算编织矩阵"""
        phase = self.anyon1.compute_braiding_phase(self.anyon2)
        
        # 2x2编织矩阵的标准形式
        elements = [
            [PhiComplex(PhiReal.one(), PhiReal.zero()), 
             PhiComplex(PhiReal.zero(), PhiReal.zero())],
            [PhiComplex(PhiReal.zero(), PhiReal.zero()), 
             PhiComplex(phase.cos(), phase.sin())]
        ]
        
        return PhiMatrix(elements=elements, dimensions=(2, 2))
    
    def compose(self, other: 'BraidingOperation') -> 'BraidingOperation':
        """复合两个编织操作"""
        # 验证复合操作仍满足no-11约束
        combined_seq = self.operation_sequence + other.operation_sequence
        
        # 合并任意子（这里简化为选择第一个操作的任意子）
        new_matrix = self.braiding_matrix.multiply(other.braiding_matrix)
        
        return BraidingOperation(
            anyon1=self.anyon1,
            anyon2=other.anyon2, 
            braiding_matrix=new_matrix,
            operation_sequence=combined_seq
        )

@dataclass
class FibonacciGate:
    """Fibonacci量子门"""
    gate_level: int         # 门层级 (0=I, 1=X, k=F_k)
    target_qubits: List[int]  # 目标量子比特
    control_qubits: List[int] = None  # 控制量子比特
    gate_matrix: PhiMatrix = None   # 门矩阵
    
    def __post_init__(self):
        """初始化门矩阵"""
        if self.gate_matrix is None:
            self.gate_matrix = self._compute_fibonacci_gate_matrix()
            
        if self.control_qubits is None:
            self.control_qubits = []
    
    def _compute_fibonacci_gate_matrix(self) -> PhiMatrix:
        """计算Fibonacci门矩阵"""
        if self.gate_level == 0:
            # F_0 = I (单位矩阵)
            return PhiMatrix.identity(2)
        elif self.gate_level == 1:
            # F_1 = X (Pauli-X门)
            elements = [
                [PhiComplex.zero(), PhiComplex.one()],
                [PhiComplex.one(), PhiComplex.zero()]
            ]
            return PhiMatrix(elements=elements, dimensions=(2, 2))
        else:
            # F_k = F_{k-1} ⊗ F_{k-2} (Fibonacci递归)
            F_prev = FibonacciGate(self.gate_level - 1, [0])._compute_fibonacci_gate_matrix()
            F_prev2 = FibonacciGate(self.gate_level - 2, [0])._compute_fibonacci_gate_matrix()
            return F_prev.tensor_product(F_prev2)
    
    def apply_to_state(self, state_vector: List[PhiComplex]) -> List[PhiComplex]:
        """将门应用到量子态"""
        # 实现量子门作用
        n_qubits = len(self.target_qubits)
        dim = 2 ** n_qubits
        
        if len(state_vector) != dim:
            raise ValueError(f"状态向量维度{len(state_vector)}与量子比特数{n_qubits}不匹配")
        
        # 应用门矩阵
        result = self.gate_matrix.matrix_vector_multiply(state_vector)
        return result
    
    def gate_complexity(self) -> int:
        """计算门复杂度"""
        return int(PHI.decimal_value ** self.gate_level)

class TopologicalQuantumComputer:
    """φ-拓扑量子计算机"""
    
    def __init__(self, n_qubits: int):
        self.phi = PHI
        self.n_qubits = n_qubits
        
        # 初始化拓扑相
        self.topological_phases: List[TopologicalPhase] = []
        self._initialize_topological_phases()
        
        # 初始化任意子系统
        self.anyons: Dict[int, Anyon] = {}
        self._initialize_anyon_system()
        
        # 量子态
        self.quantum_state = self._initialize_quantum_state()
        
        # 编织历史
        self.braiding_history: List[BraidingOperation] = []
        
        # 拓扑熵
        self.topological_entropy = PhiReal.zero()
    
    def _initialize_topological_phases(self):
        """初始化拓扑相系统"""
        for i in range(min(5, self.n_qubits)):  # 最多5个拓扑相
            fib_rank = self._fibonacci_number(i)
            
            # 计算能隙: Δ_n = Δ_0 * φ^(-n)
            energy_gap = BASIC_ENERGY_GAP * (self.phi ** (-i))
            
            # 计算相干时间: τ = τ_0 * φ^n  
            coherence_time = BASIC_COHERENCE_TIME * (self.phi ** i)
            
            phase_type = self._get_phase_type(i)
            
            phase = TopologicalPhase(
                phase_type=phase_type,
                topological_rank=fib_rank,
                energy_gap=energy_gap,
                coherence_time=coherence_time,
                phase_index=i
            )
            
            if phase.is_valid_no11():
                self.topological_phases.append(phase)
    
    def _initialize_anyon_system(self):
        """初始化任意子系统"""
        for i in range(self.n_qubits):
            # 只创建满足no-11约束的任意子标签
            if self._satisfies_no11_label(i):
                fusion_rules = self._compute_fusion_rules(i)
                topological_charge = self.phi ** (-i/2)
                
                anyon = Anyon(
                    label=i,
                    statistical_phase=None,  # 将在__post_init__中计算
                    fusion_rules=fusion_rules,
                    topological_charge=topological_charge
                )
                
                self.anyons[i] = anyon
    
    def _initialize_quantum_state(self) -> List[PhiComplex]:
        """初始化量子态为|0...0⟩"""
        dim = 2 ** self.n_qubits
        state = [PhiComplex.zero() for _ in range(dim)]
        state[0] = PhiComplex.one()  # |00...0⟩态
        return state
    
    def _fibonacci_number(self, n: int) -> int:
        """计算第n个Fibonacci数（n为输入索引）"""
        if n <= 1:
            return 1
        a, b = 1, 1
        for _ in range(n - 1):
            a, b = b, a + b
        return b
    
    def _get_phase_type(self, index: int) -> TopologicalPhaseType:
        """根据索引确定拓扑相类型"""
        if index == 0:
            return TopologicalPhaseType.TRIVIAL
        elif index == 1:
            return TopologicalPhaseType.ISING
        elif index == 2:
            return TopologicalPhaseType.FIBONACCI
        elif index == 3:
            return TopologicalPhaseType.TRIPLE
        else:
            return TopologicalPhaseType.HIGHER
    
    def _satisfies_no11_label(self, label: int) -> bool:
        """检查任意子标签是否满足no-11约束"""
        binary = format(label, 'b')
        return '11' not in binary
    
    def _compute_fusion_rules(self, label: int) -> Dict[int, PhiReal]:
        """计算融合规则 N_{ab}^c"""
        rules = {}
        
        for other_label in range(min(8, 2 * label + 1)):  # 限制搜索范围
            if self._satisfies_no11_label(other_label):
                # 融合系数遵循Fibonacci递归
                if other_label <= label:
                    coeff = self.phi ** (-(label - other_label))
                else:
                    coeff = PhiReal.zero()
                
                if coeff.decimal_value > 1e-10:
                    rules[other_label] = coeff
        
        return rules
    
    def apply_braiding(self, anyon1_label: int, anyon2_label: int, 
                      sequence: List[int]) -> None:
        """应用编织操作"""
        if anyon1_label not in self.anyons or anyon2_label not in self.anyons:
            raise ValueError("任意子标签不存在")
        
        anyon1 = self.anyons[anyon1_label]
        anyon2 = self.anyons[anyon2_label]
        
        # 验证任意子组合满足no-11约束
        if not anyon1.satisfies_no11_constraint(anyon2):
            raise ValueError("任意子组合违反no-11约束")
        
        braiding_op = BraidingOperation(
            anyon1=anyon1,
            anyon2=anyon2,
            braiding_matrix=None,  # 将在__post_init__中计算
            operation_sequence=sequence
        )
        
        # 应用编织到量子态
        self._apply_braiding_to_state(braiding_op)
        
        # 记录编织历史
        self.braiding_history.append(braiding_op)
        
        # 更新拓扑熵
        self._update_topological_entropy()
    
    def _apply_braiding_to_state(self, braiding_op: BraidingOperation):
        """将编织操作应用到量子态"""
        # 简化实现：应用编织矩阵到相关的量子比特子空间
        matrix = braiding_op.braiding_matrix
        
        # 找到受影响的量子比特索引
        affected_qubits = [braiding_op.anyon1.label % self.n_qubits,
                          braiding_op.anyon2.label % self.n_qubits]
        
        # 在子空间中应用变换
        self._apply_matrix_to_subspace(matrix, affected_qubits)
    
    def _apply_matrix_to_subspace(self, matrix: PhiMatrix, qubits: List[int]):
        """在指定量子比特子空间中应用矩阵"""
        # 这里需要实现复杂的张量积操作
        # 为了保持完整性，实现基本框架
        if len(qubits) != 2:
            raise ValueError("目前只支持两量子比特编织")
        
        # 应用矩阵变换（简化实现）
        # 实际实现需要完整的张量积计算
        pass
    
    def _update_topological_entropy(self):
        """更新拓扑熵"""
        # ΔS = k_B ln(φ) * n_anyons
        n_anyons = len([a for a in self.anyons.values() if a.topological_charge.decimal_value > 1e-10])
        entropy_increase = BOLTZMANN_CONSTANT * PHI.ln() * PhiReal.from_decimal(n_anyons)
        self.topological_entropy = self.topological_entropy + entropy_increase
    
    def apply_fibonacci_gate(self, gate_level: int, target_qubits: List[int],
                           control_qubits: List[int] = None) -> None:
        """应用Fibonacci量子门"""
        if any(q >= self.n_qubits for q in target_qubits):
            raise ValueError("目标量子比特超出范围")
        
        gate = FibonacciGate(
            gate_level=gate_level,
            target_qubits=target_qubits,
            control_qubits=control_qubits or []
        )
        
        # 检查门复杂度
        complexity = gate.gate_complexity()
        if complexity > MAX_GATE_COMPLEXITY:
            raise ValueError(f"门复杂度{complexity}超过最大限制{MAX_GATE_COMPLEXITY}")
        
        # 应用门到量子态
        if len(target_qubits) == 1:
            # 单量子比特门
            self._apply_single_qubit_gate(gate, target_qubits[0])
        else:
            # 多量子比特门
            self._apply_multi_qubit_gate(gate)
    
    def _apply_single_qubit_gate(self, gate: FibonacciGate, target: int):
        """应用单量子比特门"""
        # 实现单量子比特门的作用
        # 这需要在完整的希尔伯特空间中操作
        pass
    
    def _apply_multi_qubit_gate(self, gate: FibonacciGate):
        """应用多量子比特门"""
        # 实现多量子比特门的作用
        pass
    
    def measure_topological_charge(self, anyon_label: int) -> PhiReal:
        """测量任意子的拓扑荷"""
        if anyon_label not in self.anyons:
            raise ValueError("任意子不存在")
        
        anyon = self.anyons[anyon_label]
        return anyon.topological_charge
    
    def compute_fault_tolerance_threshold(self) -> PhiReal:
        """计算容错阈值"""
        # p_th = (φ-1)/φ
        return (self.phi - PhiReal.one()) / self.phi
    
    def get_topological_entropy(self) -> PhiReal:
        """获取当前拓扑熵"""
        return self.topological_entropy
    
    def verify_topological_protection(self) -> bool:
        """验证拓扑保护是否有效"""
        # 检查所有拓扑相的能隙
        for phase in self.topological_phases:
            if phase.energy_gap.decimal_value < MIN_ENERGY_GAP.decimal_value:
                return False
        
        # 检查编织操作的幺正性
        for braiding in self.braiding_history:
            if not self._is_unitary(braiding.braiding_matrix):
                return False
        
        return True
    
    def _is_unitary(self, matrix: PhiMatrix) -> bool:
        """检验矩阵是否幺正"""
        # 简化检验：|det(U)| = 1
        det = matrix.determinant()
        return abs(det.norm() - 1.0) < 1e-10
    
    def get_computation_statistics(self) -> Dict[str, PhiReal]:
        """获取计算统计信息"""
        stats = {}
        
        # 编织操作数
        stats["braiding_operations"] = PhiReal.from_decimal(len(self.braiding_history))
        
        # 平均相干时间
        if self.topological_phases:
            avg_coherence = sum(p.coherence_time.decimal_value for p in self.topological_phases) / len(self.topological_phases)
            stats["average_coherence_time"] = PhiReal.from_decimal(avg_coherence)
        
        # 拓扑保护度
        min_gap = min(p.energy_gap.decimal_value for p in self.topological_phases) if self.topological_phases else 0
        stats["topological_protection"] = PhiReal.from_decimal(min_gap)
        
        # 熵增率
        if self.braiding_history:
            entropy_rate = self.topological_entropy.decimal_value / len(self.braiding_history)
            stats["entropy_increase_rate"] = PhiReal.from_decimal(entropy_rate)
        
        return stats

# 物理常数和参数
PHI = PhiReal.from_decimal(1.618033988749895)
BASIC_ENERGY_GAP = PhiReal.from_decimal(1e-3)  # 基本能隙 (eV)
BASIC_COHERENCE_TIME = PhiReal.from_decimal(1e-6)  # 基本相干时间 (s)  
MIN_ENERGY_GAP = PhiReal.from_decimal(1e-6)  # 最小保护能隙 (eV)
MAX_GATE_COMPLEXITY = 1000  # 最大门复杂度
BOLTZMANN_CONSTANT = PhiReal.from_decimal(8.617333e-5)  # eV/K

# 验证函数
def verify_fibonacci_recursion(phases: List[TopologicalPhase]) -> bool:
    """验证拓扑相的Fibonacci递归关系"""
    if len(phases) < 3:
        return True
    
    for i in range(2, len(phases)):
        expected_rank = phases[i-1].topological_rank + phases[i-2].topological_rank
        if phases[i].topological_rank != expected_rank:
            return False
    
    return True

def verify_no11_constraint_in_anyons(anyons: Dict[int, Anyon]) -> bool:
    """验证任意子系统满足no-11约束"""
    labels = list(anyons.keys())
    labels.sort()
    
    for i in range(len(labels) - 1):
        if labels[i+1] - labels[i] == 1:
            # 检查相邻标签的任意子是否满足约束
            anyon1 = anyons[labels[i]]
            anyon2 = anyons[labels[i+1]]
            if not anyon1.satisfies_no11_constraint(anyon2):
                return False
    
    return True

def verify_braiding_group_structure(operations: List[BraidingOperation]) -> bool:
    """验证编织操作形成群结构"""
    # 检查群公理
    # 1. 结合律
    # 2. 单位元存在
    # 3. 逆元存在
    
    # 简化验证：检查操作序列的no-11约束
    for op in operations:
        if not op._validate_braiding_sequence():
            return False
    
    return True

验证条件

1. 拓扑自指性: T = T[T]
2. Fibonacci递归: $r_n = r_{n-1} + r_{n-2}$
3. no-11约束: 所有标签和操作序列满足
4. φ-统计相位: $θ_{ab} = 2π/φ^{|a-b|}$
5. 能隙标度: $Δ_n = Δ_0 · φ^{-n}$
6. 拓扑熵增: $dS/dt = k_B ln(φ) · n_{anyons}$